Journal of mathematics and computer science 13 (2014), 94-114 Fuzzy semi open soft sets related properties in fuzzy soft topological spaces A. Kandil a , O. A. E. Tantawy b , S. A. El-Sheikh c , A. M. Abd El-latif c a Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt. dr.ali_kandil com yahoo. @ b Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt. drosamat com yahoo. @ c Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt. sobhyelsheikh com yahoo. @ , com yahoo Alaa . 8560@ Article history: Received August 2014 Accepted September 2014 Available online September 2014 Abstract In the present paper, we continue the study on fuzzy soft topological spaces and investigate the properties of fuzzy semi open (closed) soft sets, fuzzy semi soft interior (closure), fuzzy semi continuous (open) soft functions and fuzzy semi separation axioms which are important for further research on fuzzy soft topology. In particular, we study the relationship between fuzzy semi soft interior fuzzy semi soft closure. Moreover, we study the properties of fuzzy soft semi regular spaces and fuzzy soft semi normal spaces. This paper, not only can form the theoretical basis for further applications of topology on soft sets, but also lead to the development of information systems. Keywords: Soft set, Fuzzy soft set, Fuzzy soft topological space, Fuzzy semi soft interior, Fuzzy semi soft closure, Fuzzy semi open soft, Fuzzy semi closed soft, Fuzzy semi continuous soft functions, Fuzzy soft semi separation axioms, Fuzzy soft semi i T -spaces 1,2,3,4) = (i , Fuzzy soft semi regular, Fuzzy soft semi normal. 1. Introduction The concept of soft sets was first introduced by Molodtsov [25] in 1999 as a general mathematical tool for dealing with uncertain objects. In [25, 26], Molodtsov successfully applied the soft theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory
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Journal of mathematics and computer science 13 (2014), 94-114
Fuzzy semi open soft sets related properties in fuzzy soft topological spaces
A. Kandil a , O. A. E. Tantawy b , S. A. El-Sheikh c , A. M. Abd El-latif c
a
Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt.
dr.ali_kandil comyahoo.@ b Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt.
drosamat comyahoo.@ c Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt.
sobhyelsheikh comyahoo.@ , comyahooAlaa .8560@
Article history:
Received August 2014
Accepted September 2014
Available online September 2014
Abstract In the present paper, we continue the study on fuzzy soft topological spaces and investigate the properties of fuzzy semi open (closed) soft sets, fuzzy semi soft interior (closure), fuzzy semi continuous (open) soft functions and fuzzy semi separation axioms which are important for further research on fuzzy soft topology. In particular, we study the relationship between fuzzy semi soft interior fuzzy semi soft closure. Moreover, we study the properties of fuzzy soft semi regular spaces and fuzzy soft semi normal spaces. This paper, not only can form the theoretical basis for further applications of topology on soft sets, but also lead to the development of information systems.
1. Introduction The concept of soft sets was first introduced by Molodtsov [25] in 1999 as a general mathematical tool for dealing with uncertain objects. In [25, 26], Molodtsov successfully applied the soft theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
of measurement, and so on. After presentation of the operations of soft sets [23], the properties and applications of soft set theory have been studied increasingly [4,18,26]. Xiao et al.[37] and Pei and Miao [29] discussed the relationship between soft sets and information systems. They showed that soft sets are a class of special information systems. In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets [1,6,9,13,21,22,23,24,26,27,40]. To develop soft set theory, the operations of the soft sets are redefined and a uni-int decision making method was constructed by using these new operations [10]. Recently, in 2011, Shabir and Naz [33] initiated the study of soft topological spaces. They defined soft topology on the collection of soft sets over X . Consequently, they defined basic notions of soft topological spaces such as open soft and closed soft sets, soft subspace, soft closure, soft nbd of a point, soft separation axioms, soft regular spaces and soft normal spaces and established their several properties. Min in [36] investigate some properties of these soft separation axioms. Kandil et al. [17] introduce the notion of soft semi separation axioms. In particular they study the properties of the soft semi regular spaces and soft semi normal spaces. Maji et. al. [21] initiated the study involving both fuzzy sets and soft sets. In [8], the notion of fuzzy set soft set was introduced as a fuzzy generalization of soft sets and some basic properties of fuzzy soft sets are discussed in detail. Then many scientists such as X. Yang et. al. [38], improved the concept of fuzziness of soft sets. In [1,2], Karal and Ahmed defined the notion of a mapping on classes of (fuzzy) soft sets, which is fundamental important in (fuzzy) soft set theory, to improve this work and they studied properties of (fuzzy) soft images and (fuzzy) soft inverse image s of fuzzy soft sets. Tanay et.al. [35] introduced the definition of fuzzy soft topology over a subset of the initial universe set while Roy and Samanta [32] gave the definition f fuzzy soft topology over the initial universe set. Chang [11] introduced the concept of fuzzy topology Ο on a set X by axiomatizing a collection of
fuzzy subsets of X . In the present paper, we introduce the some new concepts in fuzzy soft topological spaces such as fuzzy semi open soft sets, fuzzy semi closed soft sets, fuzzy semi soft interior, fuzzy semi soft closure and fuzzy semi separation axioms. In particular we study the relationship between fuzzy semi soft interior fuzzy semi soft closure. Also, we study the properties of fuzzy soft semi regular spaces and fuzzy soft semi normal spaces.
Moreover, we show that if every fuzzy soft point ef is fuzzy semi closed soft set in a
fuzzy soft topological space ),,( EX , then ),,( EX , is fuzzy soft semi 1T -space (resp.
fuzzy soft semi 2T -space). This paper, not only can form the theoretical basis for further
applications of topology on soft sets, but also lead to the development of information systems.
2. Preliminaries In this section, we present the basic definitions and results of soft set theory which will be needed in the sequel. For more details see [1,4,5,8,11,12,14,21,22,23,24,26,27,30,40].
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
Definition 2.1 [39] A fuzzy set A of a non-empty set X is characterized by a
membership function IXA =[0,1]: whose value )(xA represents the "degree of
membership" of x in A for Xx . Let XI denotes the family of all fuzzy sets on
X . If XIBA , , then some basic set operations for fuzzy sets are given by Zadeh [39],
as follows:
1- XxxxBA BA )()( .
2- XxxxBA BA )(=)(= .
3- XxxxxBAC BAC )()(=)(= .
4- XxxxxBAD BAD )()(=)(= .
5- XxxxAM AM )(1=)(= .
Definition 2.2 [25] Let X be an initial universe and E be a set of parameters. Let )(XP denote the power set of X and A be a non-empty subset of E . A pair
),( AF denoted by AF is called a soft set over X , where F is a mapping given by
)(: XPAF . In other words, a soft set over X is a parametrized family of subsets of
the universe X . For a particular Ae , )(eF may be considered the set of
e -approximate elements of the soft set ),( AF and if Ae , then =)(eF i.e
)}(:,:)({= XPAFEAeeFFA . The family of all these soft sets over X
denoted by AXSS )( .
Definition 2.3 [20] The union of two soft sets ),( AF and ),( BG over the common
universe X is the soft set ),( CH , where BAC = and for all Ce ,
.),()(
,),(
,),(
=)(
BAeeGeF
ABeeG
BAeeF
eH
Definition 2.4 [23] The intersection of two soft sets ),( AF and ),( BG over the
common universe X is the soft set ),( CH , where BAC = and for all Ce ,
)()(=)( eGeFeH . Note that, in order to efficiently discuss, we consider only soft sets
),( EF over a universe X with the same set of parameter E . We denote the family
of these soft sets by EXSS )( .
Definition 2.5 [33] Let be a collection of soft sets over a universe X with a fixed
set of parameters E , then EXSS )( is called a soft topology on X if;
1- ~
,~X , where =)(
~e and EeXeX ,=)(
~,
2-the union of any number of soft sets in belongs to , 3-the intersection of any two soft sets in belongs to . The triplet ),,( EX is called a soft topological space over X .
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
2- Kpupu YFSSBgBgBgff )(),(),())),((( 1 . If puf is surjective, then the equality
holds.
3- 1( , ) ( (( , ))) ( , ) ( )pu pu Ef A f f f A f A FSS X Γ΄ . If puf is injective, then the equality
holds.
4- KEpuf 0~
=)0~
( , (1 ) 1pu E Kf Γ΄ . If puf is surjective, then the equality holds.
5- EKpuf 1~
=)1~
(1 and EKpuf 0~
=)0~
(1 .
6- If ),(),( AgAf , then
),(),( AgfAff pupu
.
7- If ( , ) ( , )f B g BΓ΄ , then
1 1( , ) ( , ) ( , ), ( , ) ( ) .pu pu Kf f B f g B f B g B FSS Y Γ΄
8- jpuJjjJjpu BffBff ),(=)),(( 11
and
KjjpuJjjJjpu YFSSBfBffBff )(),(,),(=)),(( 11
.
9- jpuJjjJjpu AffAff ),(=)),(( and
EjjpuJjjJjpu XFSSAfAffAff )(),(),()),(( . If puf is injective, then the
equality holds.
Definition 2.22 [16] Let ),,( EX be a soft topological space and EA XSSF )( . If
))((~AA FintclF , then AF is called semi open soft set. We denote the set of all semi
open soft sets by ),,( EXSOS , or )(XSOS and the set of all semi closed soft sets by
),,( EXSCS , or )(XSCS .
3 . Fuzzy semi open (closed) soft sets Various generalization of closed and open soft sets in soft topological spaces were studied by Kandil et al. [16], but for fuzzy soft topological spaces such generalization have not been studied so far. In this section, we move one step forward to introduce fuzzy semi open and fuzzy semi closed soft sets and study various properties and notions related to these structures.
Definition 3.1 Let ),,( EX T be a fuzzy soft topological space and EA XFSSf )( . If
))(( AA fFintFclf , then Af is called fuzzy semi open soft set. We denote the set of all
fuzzy semi open soft sets by ),,( EXFSOS T , or )(XFSOS and the set of all fuzzy
semi closed soft sets by ),,( EXFSCS T , or )(XFSCS .
Theorem 3.1 Let ),,( EX T be a fuzzy soft topological space and )(XFSOSfA .Then
1- Arbitrary fuzzy soft union of fuzzy semi open soft sets is fuzzy semi open soft. 2-Arbitrary fuzzy soft intersection of fuzzy semi closed soft sets is fuzzy semi closed soft.
Proof. 1- Let )(}:),{( XFSOSJjAf j . Then Jj , .))),(((),( jj AfFclFintAf
3- )( AfFSint is the largest fuzzy semi open soft set contained in Af .
4- If A Bf gΓ΄ , then ( ) ( )A BFSint f FSint gΓ΄ .
5- )(=))(( AA fFSintfFSintFSint .
6- )]()[()()( BABA gfFSintgFSintfFSint .
7- )()()]()[( BABA gFSintfFSintgfFSint .
Proof. Obvious.
Theorem 3.5 Let ),,( EX T be a fuzzy soft topological space and EBA XFSSgf )(, .
Then the following properties are satisfied for the fuzzy semi closure operator, denoted by FScl .
1- EEFScl 1~
=)1~
( and EEFScl 0~
=)0~
( .
2- )()( AA fFSclf .
3- )( AfFScl is the smallest fuzzy semi closed soft set contains Af .
4- If BA gf , then )()( BA gFSclfFScl .
5- )(=))(( AA fFSclfFSclFScl .
6- )]()[()()( BABA gfFSclgFSclfFScl .
7- )()()]()[( BABA gFSclfFSclgfFScl .
Proof. Immediate.
Remark 3.2 Note that the family of all fuzzy semi open soft sets on a fuzzy soft topological space ),,( EX T forms a fuzzy supra soft topology, which is a collection of
fuzzy soft sets contains EE 0~
,1~
and closed under arbitrary fuzzy soft union.
Theorem 3.6 Every fuzzy open (resp. closed) soft set in a fuzzy soft topological space ),,( EX T is fuzzy semi open (resp. fuzzy semi closed) soft.
Proof. Let )(XFOSfA . Then AA ffFint =)( . Since )( AA fFclf , then
))(( AA fFintFclf . Thus, )(XFSOSfA .
Remark 3.3 The converse of Theorem 3.6 is not true in general as shown in the following example.
Example 3.1 Let },,{= cbaX , },,{= 321 eeeE and EDCBA ,,, where },{= 21 eeA ,
},{= 32 eeB , },{= 31 eeC and }{= 2eD . Let },,,,,,0~
,1~
{= 654321 DBEDBAEE ffffffT
where DBEDBA ffffff 654321 ,,,,, are fuzzy soft sets over X defined as follows:
},,{= 0.40.750.51
1cba
e
Af , },,{= 0.70.80.3
2
1cba
e
Af ,
},,{= 0.30.60.42
2cba
e
Bf , },,{= 0.450.40.2
3
2cba
e
Bf ,
},,{= 0.30.60.32
3cba
e
Df ,
},,{= 0.40.750.51
4cba
e
Ef , },,{= 0.70.80.4
2
4cba
e
Ef , },,{= 0 . 4 50 . 40 . 2
3
4cba
e
Ef ,
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
Theorem 3.10 Let ),,( EX T be a fuzzy soft topological space and EA XFSSf )( .
Then )(XFSCSfA if and only if AA ffFclFint ))(( .
Proof. Let )(XFSCSfA , then Af is fuzzy semi open soft set. This means that,
)))(((1~
=))1~
(( AEAEA fFclFintfFintFclf . Therefore, AA ffFclFint ))(( .
Conversely, let AA ffFclFint Γ΄))(( . Then ))1~
((1~
AEAE fFintFclf Γ΄ . Hence, AE f1~
is
fuzzy semi open soft set. Therefore, Af is fuzzy semi closed soft set.
Corollary 3.1 Let ),,( EX T be a fuzzy soft topological space and EA XFSSf )( . Then
)(XFSCSfA if and only if ))((= AAA fFclFintff .
Proof. It is obvious from Theorem 3.10.
4. Fuzzy semi continuous soft functions Kharal et al. [1,2] introduced soft function over classes of (fuzzy) soft sets. The authors also defined and studied the properties of soft images and soft inverse images of (fuzzy) soft sets, and used these notions to the problem of medical diagnosis in medical expert systems. Kandil et al. [17] introduced some types of soft function in soft topological spaces. Here we introduce the notions of fuzzy semi soft function in fuzzy soft topological spaces and study its basic properties.
Definition 4.1 Let ),,( 1 EX T , ),,( 2 KY T be fuzzy soft topological spaces and
KEpu YFSSXFSSf )()(: be a soft function. Then puf is called;
1- Fuzzy semi continuous soft function if 2
1 )()( T
BBpu gXFSOSgf .
2- Fuzzy fuzzy semi open soft if 1)()( T AApu gYFSOSgf .
3- Fuzzy semi closed soft if 1')()( T AApu fYFSCSff .
4- Fuzzy irresolute soft if )()()(1 YFSOSgXFSOSgf BBpu .
5- Fuzzy irresolute open soft if )()()( XFSOSgYFSOSgf AApu .
6- Fuzzy irresolute closed soft if )()()( YFSCSfYFSCSff AApu .
Example 4.1 Let },,{== cbaYX , },,{= 321 eeeE and EA where },{= 21 eeA . Let
),,(),,(: 21 KYEXf pu TT be the constant soft mapping where 1T is the indiscrete
fuzzy soft topology and 2T is the discrete fuzzy soft topology such that
Xxaxu =)( and Eeeep 1=)( . Let Af be fuzzy soft sets over Y defined as
follows:
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
Proof. Let KD YFSSk )( and 1'TCl such that CDpu lkf )(1 . Then, DCpu klf )(
from Theorem 2.3 where 1TCl . Since puf is fuzzy semi open soft function. Then
)()( YFSOSlf Cpu . Take ])([= CpuB lfh . Hence )(YFSCShB such that BD hk and
CDpuDpuCpupuBpu lkfkflffhf )(=)()])(([=)( 1111 . This completes the proof.
Theorem 4.5 Let ),,( 1 EX T and ),,( 2 KY T be fuzzy soft topological spaces and puf
be a soft function such that KEpu YFSSXFSSf )()(: . Then the following are
equivalent:
1- puf is fuzzy semi closed soft function.
2- EAApuApu XFSShhFclfhfFScl )())(())((1
T .
Proof.
π β π Let 1'TAh . Then 1')()( T AApu fYFSCShf by (1). Hence,
EAApuApuApu XFSShhFclfhfhfFScl )())(()(=))((1
T .
π β π Let 1'TAg . By hypothesis, )(=))(())((1
ApuApuApu hfhFclfhfFScl T . Hence,
1')()( T AApu hYFSCShf . Therefore, puf is fuzzy semi closed soft function.
5. Fuzzy soft semi separation axioms Soft separation axioms for soft topological spaces were studied by Shabir and Naz [33]. Kandil et al. [17] introduced the notions of soft semi separation axioms in soft topological spaces. Here we introduce the notions of fuzzy semi connectedness in fuzzy soft topological spaces and study its basic properties.
Definition 5.1 Two fuzzy soft points ee xf = and
ee yg = are said
to be distinct if and only if π₯ β π¦.
Definition 5.2 A fuzzy soft topological space ),,( EX T is said to be a fuzzy soft semi
oT -space if for every pair of distinct fuzzy soft points ee gf , there exists a fuzzy semi
open soft set containing one but not the other.
Examples 5.1
1- Let },{= baX , },{= 21 eeE and T be the discrete fuzzy soft topology on X .
Then ),,( EX T is fuzzy soft semi oT -space.
2- Let },{= baX , },{= 21 eeE and T be the indiscrete fuzzy soft topology on X .
Then T is not fuzzy soft semi oT -space.
Theorem 5.1 A soft subspace ),,( EY YT of a fuzzy soft semi oT -space ),,( EX T is
fuzzy soft semi oT .
Proof. Let ee gh , be two distinct fuzzy soft points of YT . Then these fuzzy soft points
A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif/ J. Math. Computer Sci. 13 (2014),94-114
irresolute open soft function. Then )(),( DpuCpu hfkf are fuzzy semi open soft sets in Y
. Thus, ),,( 2 KY T is fuzzy soft semi normal space.
6 . Conclusion Topology is an important and major area of mathematics and it can give many relationships between other scientific areas and mathematical models. Recently, many scientists have studied the soft set theory, which is initiated by Molodtsov [25] and easily applied to many problems having uncertainties from social life. In the present work, we have continued to study the properties of fuzzy soft topological spaces. We introduce the some new concepts in fuzzy soft topological spaces such as fuzzy semi open soft sets, fuzzy semi closed soft sets, fuzzy semi soft interior, fuzzy semi soft closure and fuzzy semi separation axioms and have established several interesting properties. Since the authors introduced topological structures on fuzzy soft sets [8,15, 35], so the semi topological properties, which introduced by Kandil et al.[17], is generalized here to the fuzzy soft sets which will be useful in the fuzzy systems. Because there exists compact connections between soft sets and information systems [29,37], we can use the results deducted from the studies on fuzzy soft topological space to improve these kinds of connections. We hope that the findings in this paper will help researcher enhance and promote the further study on fuzzy soft topology to carry out a general framework for their applications in practical life.
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